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Over 100 years, Methane has a Global warming potential 34 times bigger than Carbon dioxide. Over 20 years, it's even 86 times bigger. How does one arrive at these numbers?

Looking at the following graphic doesn't help either, as the area for Methane barely covers the left margin of the upgoing thermal radiation and Carbon dioxide is at the right margin. The area under the absorption curve for Carbon dioxide is definitely bigger than the one for Methane.

Atmospheric Transmission

Source: https://commons.wikimedia.org/wiki/File:Atmospheric_Transmission.png

So what makes Methane a more effective greenhouse gas?

integralette
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Global warming potential is given per mole, but you plot the atmospheric column-integrated effect.
Methane concentrations are about 2 ppm in the atmosphere, that of $\rm CO_2$ is 400 ppm, still methane manages to make a visible blip in the total atmospheric absorption plot.

As to any molecules "why is X a GHG, but Y is not?", the answer is in short "the positions and strength of the molecular absorption bands relative to the terrestrial infrared emission gives your greenhouse warming potential per mole", or in more words here on earth-science.

AtmosphericPrisonEscape
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